To determine the sound produced by turbulence near an elastic boundary, it is necessary to know the response of the boundary to the turbulence stresses. These stresses not only generate sound but also excite structural vibrations that can store a significant amount of flow energy. The vibrations are ultimately dissipated by frictional forces, but they can contribute substantially to the radiated noise because elastic waves are “scattered” at structural discontinuities, and some of their energy is transformed into sound. Thus, flow-generated sound reaches the far field via two paths: directly from the turbulence sources and indirectly from possibly remote locations where the scattering occurs. The result is that the effective acoustic efficiency of the flow can be very much larger than for a geometrically similar rigid surface, even when only a small fraction of the structural energy is scattered into sound. Typical examples include the cabin noise produced by turbulent flow over an aircraft fuselage and the noise radiated from ship and submarine hulls, from duct flows, piping systems, and turbomachines [26]. Interactions of this kind are discussed in this chapter.

Sources Near an Elastic Plate

The simplest flexible boundary is the homogeneous, nominally flat, thin elastic plate, which supports structural modes in the form of bending waves. The effects of fluid loading are usually important in liquids, where the Mach number M is small, and in this section, it will be assumed that M ≪ 1 and, therefore, that mean flow has a negligible effect on the propagation of sound and plate vibrations.

Aerodynamic Sound

The sound generated by vorticity in an unbounded fluid is called aerodynamic sound [60, 61]. Most unsteady flows of technological interest are of high Reynolds number and turbulent, and the acoustic radiation is a very small by-product of the motion. The turbulence is usually produced by fluid motion relative to solid boundaries or by the instability of free shear layers separating a high-speed flow (such as a jet) from a stationary environment. In this chapter the influence of boundaries on the production of sound as opposed to the production of vorticity will be ignored. The aerodynamic sound problem then reduces to the study of mechanisms that convert kinetic energy of rotational motions into acoustic waves involving longitudinal vibrations of fluid particles. There are two principal source types in free vortical flows: a quadrupole, whose strength is determined by the unsteady Reynolds stress, and a dipole, which is important when mean mass density variations occur within the source region.

Lighthill's Acoustic Analogy

The theory of aerodynamic sound was developed by Lighthill [60], who reformulated the Navier–Stokes equation into an exact, inhomogeneous wave equation whose source terms are important only within the turbulent (vortical) region. Sound is expected to be such a very small component of the whole motion that, once generated, its back-reaction on the main flow is usually negligible. In a first approximation the motion in the source region may then be determined by neglecting the production and propagation of the sound.

Influence of Rigid Boundaries on the Generation of Aerodynamic Sound

The Ffowcs Williams–Hawkings equation (2.2.3) enables aerodynamic sound to be represented as the sum of the sound produced by the aerodynamic sources in unbounded flow together with contributions from monopole and dipole sources distributed on boundaries. For turbulent flow near a fixed rigid surface, the direct sound from the quadrupoles Tij is augmented by radiation from surface dipoles whose strength is the force per unit surface area exerted on the fluid. If the surface is in accelerated motion, there are additional dipoles and quadrupoles, and neighboring surfaces in relative motion also experience “potential flow” interactions that generate sound. At low Mach numbers, M, the acoustic efficiency of the surface dipoles exceeds the efficiency of the volume quadrupoles by a large factor ∼O(1/M2) (Sections 1.8 and 2.1). Thus, the presence of solid surfaces within low Mach number turbulence can lead to substantial increases in aerodynamic sound levels. Many of these interactions are amenable to precise analytical modeling and will occupy much of the discussion in this chapter.

Acoustically Compact Bodies [70]

Consider the production of sound by turbulence near a compact, stationary rigid body. Let the fluid have uniform mean density p0 and sound speed c0, and assume the Mach number is sufficiently small that convection of the sound by the flow may be neglected. This particular situation arises frequently in applications. In particular, M rarely exceeds about 0.01 in water, and sound generation by turbulence is usually negligible except where the flow interacts with a solid boundary [111].

Jets and shear layers are frequently responsible for the generation of intense acoustic tones. Instability of the mean flow over of a wall cavity excites “self-sustained” resonant cavity modes or periodic “hydrodynamic” oscillations, which are maintained by the steady extraction of energy from the flow. Whistles and musical instruments such as the flute and organ pipe are driven by unstable air jets, and shear layer instabilities are responsible for tonal resonances excited in wind tunnels, branched ducting systems, and in exposed openings on ships and aircraft and other high-speed vehicles. These mechanisms are examined in this chapter, starting with very high Reynolds number flows, where a shear layer can be approximated by a vortex sheet. We shall also discuss resonances where thermal processes play a fundamental role, such as in the Rijke tube and pulsed combustor. The problems to be investigated are generally too complicated to be treated analytically with full generality, but much insight can be gained from exact treatments of linearized models and by approximate nonlinear analyses based on simplified, yet plausible representations of the flow.

Linear Theory of Wall Aperture and Cavity Resonances

Stability of Flow Over a Circular Wall Aperture

The sound produced by nominally steady, high Reynolds number flow over an opening in a thin wall is the simplest possible system to treat analytically. Our approach is applicable to all linearly excited systems involving an unstable shear layer, and it is an extension of the method used in Section 5.3.6 to determine the conductivity of a circular aperture in a mean grazing flow.

Fluid motion in the immediate vicinity of a solid surface is usually controlled by viscous stresses that cause an adjustment in the velocity to comply with the no-slip condition and by thermal gradients that similarly bring the temperatures of the solid and fluid to equality at the surface. At high Reynolds numbers, these adjustments occur across boundary layers whose thicknesses are much smaller than the other governing length scales of the motion. We have seen how the forced production of vorticity in boundary layers during convection of a “gust” past the edge of an airfoil can be modeled by application of the Kutta condition (Section 3.3). In this chapter, similar problems are discussed involving the generation of vorticity by sound impinging on both smooth surfaces and surfaces with sharp edges in the presence of flow. The aerodynamic sound generated by this vorticity augments the sound diffracted in the usual way by the surface. However, the near field kinetic energy of the vorticity is frequently derived wholly from the incident sound, so that unless the subsequent vortex motion is unstably coupled to the mean flow (acquiring additional kinetic energy from the mean stream as it evolves) there will usually be an overall decrease in the acoustic energy: The sound will be damped. General problems of this kind, including the influence of surface vibrations, are the subject of this chapter. We start with the simplest case of sound impinging on a plane wall.

Damping of Sound at a Smooth Wall

Dissipation in the Absence of Flow

The thermo-viscous attenuation of sound is greatly increased in the neighborhood of a solid boundary where temperature and velocity gradients are large.

This book deals with that branch of fluid mechanics concerned with the production and absorption of sound occuring when unsteady flow interacts with solid bodies. Problems of this kind are commonly known under the heading of aerodynamic sound but often include more conventional areas of acoustics and structural vibration. Acoustics is here regarded as a branch of fluid mechanics, and an attempt has therefore been made in Chapter 1 to provide the necessary background material in this subject. Elementary concepts of classical acoustics and structural vibrations are also reviewed in this chapter. Constraints of space and time have required the omission or the curtailed discussion of several important subareas of the acoustics of fluid-structure interactions, including in particular many problems involving supersonic flow. The book should be of value in one or more of the following ways: (i) as a reference for analytical methods for modeling acoustic problems; (ii) as a repository of known results and methods in the theory of aerodynamic sound and vibration, which have tended to become scattered throughout many journal and review articles over the past forty or so years; and (iii) as a graduate level textbook. Chapter 1 and selected topics from Chapters 2 and 3 have been used for several years in teaching an advanced graduate level course on the theory of acoustics and aerodynamic sound.

Theoretical concepts are illustrated and sometimes extended by numerous examples, many of which include complete worked solutions. Every effort has been made to ensure the accuracy of formulae, both in the main text and in the examples. The author would welcome notification of errors detected by the reader and more general suggestions for improvements.

The theory of water waves has been a source of intriguing – and often difficult – mathematical problems for at least 150 years. Virtually every classical mathematical technique appears somewhere within its confines; in addition, linear problems provide a useful exemplar for simple descriptions of wave propagation, with nonlinearity adding an important level of complexity. It is, perhaps, the most readily accessible branch of applied mathematics, which is the first step beyond classical particle mechanics. It embodies the equations of fluid mechanics, the concepts of wave propagation, and the critically important rôle of boundary conditions. Furthermore, the results of a calculation provide a description that can be tested whenever an expanse of water is to hand: a river or pond, the ocean, or simply the household bath or sink. Indeed, the driving force for many workers who study water waves is to obtain information that will help to tame this most beautiful, and sometimes destructive, aspect of nature. (Perhaps ‘to tame’ is far too bold an ambition: at least to try to make best use of our knowledge in the design of man-made structures.) Here, though, we shall – without apology – restrict our discussion to the many and varied aspects of water-wave theory that are essentially mathematical. Such studies provide an excellent vehicle for the introduction of the modern approach to applied mathematics: complete governing equations; nondimensionalisation and scaling; rational approximation; solution; interpretation. This will be the type of systematic approach that is adopted throughout this text.

Before we commence our presentation of the theory of water waves, we require a firm and precise base from which to start. This must be, at the very least, a statement of the relevant governing equations and boundary conditions. However, it is more satisfactory, we believe, to provide some background to these equations, albeit within the confines of an introductory and relatively brief chapter. The intention is therefore to present a derivation of the equations for inviscid fluid mechanics (Euler's equation and the equation of mass conservation) and a few of their properties. (The corresponding equations for a viscous fluid – primarily the Navier–Stokes equation – appear in Appendix A.) Coupled to these general equations is the set of boundary (and initial) conditions which select the water-wave problem from all other possible solutions of the equations. Of particular importance, as we shall see, are the conditions that define and describe the surface of the fluid; these include the kinematic condition and the rôles of pressure and surface tension. Some rather general consequences of coupling the equations and boundary conditions will also be mentioned.

Once we have available the complete prescription of the water-wave problem, based on a particular model (such as for inviscid flow), we may ‘normalise’ in any manner that is appropriate.

In Chapter 2 we presented some classical ideas in the theory of water waves. One particular concept that we introduced was the phenomenon of a balance between nonlinearity and dispersion, leading to the existence of the solitary wave, for example. Further, under suitable assumptions, this wave can be approximated by the sech2 function, which is an exact solution of the Korteweg–de Vries (KdV) equation; see Section 2.9.1. We shall now use this result as the starting point for a discussion of the equations, and of the properties of corresponding solutions, that arise when we invoke the assumptions of small amplitude and long wave-length. In the modern theories of nonlinear wave propagation – and certainly not restricted only to water waves – this has proved to be an exceptionally fruitful area of study.

The results that have been obtained, and the mathematical techniques that have been developed, have led to altogether novel, important and deep concepts in the theory of wave propagation. Starting from the general method of solution for the initial value problem for the KdV equation, a vast arena of equations, solutions and mathematical ideas has evolved. At the heart of this panoply is the soliton, which has caused much excitement in the mathematical and physical communities over the last 30 years or so. It is our intention to describe some of these results, and their relevance to the theory of water waves, where, indeed, they first arose.